Download Sequences Derived from a Set Types of Sequences from a Set

CSCI 1900
Discrete Structures
Permutations
Reading: Kolman, Section 3.1
CSCI 1900 – Discrete Structures
Permutations – Page 1
Types of Sequences from a Set
There are a number of different ways to create
a sequence from a set
– Any order, duplicates allowed
– Any order, no duplicates allowed
– Order matters, duplicates allowed
– Order matters, no duplicates allowed
CSCI 1900 – Discrete Structures
Permutations – Page 3
Multiplication Principle of Counting
• The first type of sequence we will look at is
where duplicates are allowed and their
order matters.
• Supposed that two tasks T1 and T2 must
be performed in sequence.
• If T1 can be performed in n1 ways, and for
each of these ways, T2 can be performed
in n2 ways, then the sequence T1T2 can be
performed in n1⋅n2 ways.
CSCI 1900 – Discrete Structures
Permutations – Page 5
Sequences Derived from a Set
• Assume we have a set A containing n
items.
• Examples include alphabet, decimal digits,
playing cards, etc.
• We can produce sequences from each of
these sets.
CSCI 1900 – Discrete Structures
Permutations – Page 2
Classifying Real-World Sequences
Determine size of set A and classify each of
the following as one of the previously listed
types of sequences
•Five card stud poker •Windows XP CD
Key
•Phone numbers
•Votes in a
•License plates
presidential election
•Lotto numbers
•Codes for 5-digit
•Binary numbers
CSCI door locks
CSCI 1900 – Discrete Structures
Permutations – Page 4
Multiplication Principle (continued)
• Extended previous example to T1, T2, …,
Tk
• Solution becomes n1⋅n2⋅…⋅nk
CSCI 1900 – Discrete Structures
Permutations – Page 6
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Examples of Multiplication Principle
• 8 character passwords
– First digit must be a letter
– Any character after that can be a letter or a
number
– 26*36*36*36*36*36*36*36 =
2,037,468,266,496
• Windows XP/2000 software keys
– 25 characters of letters or numbers
– 3625
CSCI 1900 – Discrete Structures
Permutations – Page 7
Calculation of the Number of Subsets
• Let A be a set with n elements: how many subsets
does A have?
• Each element may either be included or not included.
• In section 1.3, we talked about the characteristic
function which defines membership in a set based on
a universal set
• Example:
– U={1,2,3,4,5,6}
– A = {1,2}, B={2, 4, 6}
– fA = {1,1,0,0,0,0}, fB = {0,1,0,1,0,1}
CSCI 1900 – Discrete Structures
Permutations – Page 9
• License plates of the form “ABC 123”:
– 26*26*26*10*10*10 = 17,576,000
• Phone numbers
– Three digit area code cannot begin with 0
– Three digit exchange cannot begin with 0
– 9*10*10*9*10*10*10*10*10*10 =
8,100,000,000
CSCI 1900 – Discrete Structures
Permutations – Page 8
Calculation of the Number of
Subsets (continued)
• Every subset of A can be defined with a
characteristic function of n elements where
each element is a 1 or a 0, i.e., each
element has 2 possible values
• Therefore, there are 2 ⋅ 2 ⋅ 2 ⋅ … ⋅ 2 = 2n
possible characteristic functions
CSCI 1900 – Discrete Structures
Permutations – Page 10
Multiplication Principle Versus
Permutations
Permutations
• The next type of sequence we will look at
is where duplicates are not allowed and
their order matters
• Assume A is a set of n elements
• Suppose we want to make a sequence, S,
of length r where 1 < r < n
CSCI 1900 – Discrete Structures
More Examples of
Multiplication Principle
Permutations – Page 11
• If repeated elements are allowed, how
many different sequences can we make?
• Process:
– Each time we select an element for the next
element in the sequence, S, we have n to
choose from
– This gives us n ⋅ n ⋅ n ⋅ … ⋅ n = nr possible
choices
CSCI 1900 – Discrete Structures
Permutations – Page 12
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Multiplication Principle Versus
Permutations (continued)
• Suppose repeated elements are not allowed,
how many different sequences can we make?
• Process:
– The first selection, T1, provides n choices.
– Each time we select an element after that, Tk
where k>1, there is one less than there was for the
previous selection, k-1.
– The last choice, Tr, has n – (r – 1) = n – r + 1
choices
– This gives us n⋅(n – 1)⋅(n – 2)⋅…⋅(n – r + 1)
CSCI 1900 – Discrete Structures
Permutations – Page 13
• For r=n,
nPn = nPr = n⋅(n – 1)⋅(n – 2)⋅…⋅2⋅1
• This number is also written as n! and is
read n factorial
• nPr can be written in terms of factorials
= n⋅(n – 1)⋅(n – 2)⋅…⋅(n – r + 1)
n⋅(n – 1)⋅(n – 2)⋅…⋅(n – r + 1)⋅(n – r) ⋅…⋅2⋅1
nPr =
(n – r) ⋅…⋅2⋅1
nPr = n!/(n – r)!
CSCI 1900 – Discrete Structures
• Notation: nPr is called number of permutations
of n objects taken r at a time.
• Word scramble: How many 4 letter words can
be made from the letters in “Gilbreath” without
duplicate letters?
9P4 = 9⋅8⋅7⋅6 = 3,024
• Example, how many 4-digit PINs can be
created for the 5 button CSCI door locks?
5P4 = 5⋅4⋅3⋅2 = 120
• Would adding a fifth digit give us more PINs?
CSCI 1900 – Discrete Structures
Permutations – Page 15
• If the set from which a sequence is
being derived has duplicate elements,
e.g., {a, b, d, d, g, h, r, r, r, s, t}, then
straight permutations will actually count
some sequences multiple times.
• Example: How many words can be made
from the letters in Tarnoff?
• Problem: the f’s cannot be distinguished, e.g.,
aorf cannot be distinguished from aorf
CSCI 1900 – Discrete Structures
Distinguishable Permutations from a
Set with Repeated Elements
• Number of distinguishable permutations
that can be formed from a collection of n
objects where the first object appears k1
times, the second object k2 times, and so
on is:
n! / (k1!⋅ k2!⋅⋅⋅ kt!)
where k1 + k2 + … + kt = n
CSCI 1900 – Discrete Structures
Permutations – Page 14
Distinguishable Permutations from
a Set with Repeated Elements
Factorial
nPr
Permutations
Permutations – Page 17
Permutations – Page 16
Example
• How many distinguishable words can be
formed from the letters of JEFF?
• Solution: n = 4, kj = 1, ke = 1, kf = 2
n!/(kj! ⋅ ke! ⋅ kf!) = 4!/(1! 1! 2!) = 12
• List:
JEFF, JFEF, JFFE, EJFF, EFJF, EFFJ,
FJEF, FEJF, FJFE, FEFJ, FFJE, and
FFEJ
CSCI 1900 – Discrete Structures
Permutations – Page 18
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In-Class Exercises
Example
• How many distinguishable words can be
formed from the letters of MISSISSIPPI?
• Solution:
n = 11, km = 1, ki = 4, ks = 4, kp = 2
• How many ways can you sort a deck of 52
cards?
• Compute the number of 4-digit ATM PINs
where duplicate digits are allowed.
• How many ways can the letters in the word
“TARNOFF” be arranged?
n!/(km! ⋅ ki! ⋅ ks! ⋅ kp!) = 11!/(1! 4! 4! 2!)
= 34,650
CSCI 1900 – Discrete Structures
Permutations – Page 19
CSCI 1900 – Discrete Structures
Permutations – Page 20
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